\newproblem{lay:1_2_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.2.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  In the following equation system
	\begin{center}
		$x_1+hx_2=2$\\
		$4x_1+8x_2=k$
	\end{center}
	choose values for $h$ and $k$ such that it has (a) no solution, (b) a unique solution, and (c) many solutions.
}
{
   % Solution
	The augmented matrix of the equation system is
	\begin{center}
		$\begin{pmatrix} 1 & h & 2 \\ 4 & 8 & k \end{pmatrix}$
	\end{center}
	Let's reduce it
	\begin{center}
		\begin{tabular}{lc}
			$\mathbf{r}_2\leftarrow \mathbf{r}_2-4\mathbf{r}_1$ &
			$\begin{pmatrix} 1 & h & 2 \\ 0 & 8-4h & k-8 \end{pmatrix}$
		\end{tabular}
	\end{center}
	\begin{enumerate}[a]
		\item If $8-4h=0$ and $k-8\neq 0$, then the equation system has no solution. Two specific values are $h=2$ and $k=0$.
		\item If $8-4h\neq 0$, then there is a unique solution. In particular, for $h=k=0$, the equation system has a unique solution.
		\item If $8-4h=0$ and $k-8=0$, there are infinite solutions. Particularly, this happens for $h=2$ and $k=8$.
	\end{enumerate}
}
\useproblem{lay:1_2_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
